Solve for $x$ : $ 8|x - 9| - 10 = -4|x - 9| + 4 $
Explanation: Add $ {4|x - 9|} $ to both sides: $ \begin{eqnarray} 8|x - 9| - 10 &=& -4|x - 9| + 4 \\ \\ { + 4|x - 9|} && { + 4|x - 9|} \\ \\ 12|x - 9| - 10 &=& 4 \end{eqnarray} $ Add ${10}$ to both sides: $ \begin{eqnarray} 12|x - 9| - 10 &=& 4 \\ \\ { + 10} &=& { + 10} \\ \\ 12|x - 9| &=& 14 \end{eqnarray} $ Divide both sides by ${12}$ $ \dfrac{12|x - 9|} {{12}} = \dfrac{14} {{12}} $ Simplify: $ |x - 9| = \dfrac{7}{6}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 9 = -\dfrac{7}{6} $ or $ x - 9 = \dfrac{7}{6} $ Solve for the solution where $x - 9$ is negative: $ x - 9 = -\dfrac{7}{6} $ Add ${9}$ to both sides: $ \begin{eqnarray} x - 9 &=& -\dfrac{7}{6} \\ \\ {+ 9} && {+ 9} \\ \\ x &=& -\dfrac{7}{6} + 9 \end{eqnarray} $ Change the ${ + 9}$ to an equivalent fraction with a denominator of $6$ $ x = - \dfrac{7}{6} {+ \dfrac{54}{6}} $ $ x = \dfrac{47}{6} $ Then calculate the solution where $x - 9$ is positive: $ x - 9 = \dfrac{7}{6} $ Add ${9}$ to both sides: $ \begin{eqnarray} x - 9 &=& \dfrac{7}{6} \\ \\ {+ 9} && {+ 9} \\ \\ x &=& \dfrac{7}{6} + 9 \end{eqnarray} $ Change the ${ + 9}$ to an equivalent fraction with a denominator of $6$ $ x = \dfrac{7}{6} {+ \dfrac{54}{6}} $ $ x = \dfrac{61}{6} $ Thus, the correct answer is $x = \dfrac{47}{6} $ or $x = \dfrac{61}{6} $.